Concept in probability theory
HypoexponentialParameters |
rates (real) |
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Support |
 |
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PDF |
Expressed as a phase-type distribution
 Has no other simple form; see article for details |
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CDF |
Expressed as a phase-type distribution
 |
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Mean |
 |
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Median |
General closed form does not exist[1] |
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Mode |
if , for all k |
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Variance |
 |
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Skewness |
 |
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Ex. kurtosis |
no simple closed form |
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MGF |
 |
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CF |
 |
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In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.
Overview
The Erlang distribution is a series of k exponential distributions all with rate
. The hypoexponential is a series of k exponential distributions each with their own rate
, the rate of the
exponential distribution. If we have k independently distributed exponential random variables
, then the random variable,

is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of
.
Relation to the phase-type distribution
As a result of the definition it is easier to consider this distribution as a special case of the phase-type distribution. The phase-type distribution is the time to absorption of a finite state Markov process. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from state i to i+1 with rate
until state k transitions with rate
to the absorbing state k+1. This can be written in the form of a subgenerator matrix,
![\left[{\begin{matrix}-\lambda _((1))&\lambda _((1))&0&\dots &0&0\\0&-\lambda _((2))&\lambda _((2))&\ddots &0&0\\\vdots &\ddots &\ddots &\ddots &\ddots &\vdots \\0&0&\ddots &-\lambda _((k-2))&\lambda _((k-2))&0\\0&0&\dots &0&-\lambda _((k-1))&\lambda _((k-1))\\0&0&\dots &0&0&-\lambda _((k))\end{matrix))\right]\;.](https://wikimedia.org/api/rest_v1/media/math/render/svg/5adc28a5ad374c83a57976ffcd0f726ebea058a3)
For simplicity denote the above matrix
. If the probability of starting in each of the k states is

then
Characterization
A random variable
has cumulative distribution function given by,

and density function,

where
is a column vector of ones of the size k and
is the matrix exponential of A. When
for all
, the density function can be written as

where
are the Lagrange basis polynomials associated with the points
.
The distribution has Laplace transform of

Which can be used to find moments,
![E[X^((n))]=(-1)^((n))n!{\boldsymbol {\alpha ))\Theta ^((-n)){\boldsymbol {1))\;.](https://wikimedia.org/api/rest_v1/media/math/render/svg/77583344b090d9c7bc68e36bb0387108c41702e4)
General case
In the general case
where there are
distinct sums of exponential distributions
with rates
and a number of terms in each
sum equals to
respectively. The cumulative
distribution function for
is given by

with

with the additional convention
.
Uses
This distribution has been used in population genetics,[3] cell biology,[4][5] and queuing theory[6][7]